TMLE step by step

1 Introduction

1. TMLE is a general algorithm for the construction of double robust, semiparametric, efficient substitution estimators. TMLE allows for data-adaptive estimation while obtaining valid statistical inference.
2. Although TMLE implemtation uses the G-computation estimand (G-formula). Briefly, the TMLE algorithm uses information in the estimated exposure mechanism P(A|W) to update the initial estimator of the conditional mean E\(_{0}\)(Y|A,W).
3. The targeted estimates are then substituted into the parameter mapping. The updating step achieves a targeted bias reduction for the parameter of interest \(\psi(P_{0})\) (the true target parameter) and serves to solve the efficient score equation. As a result, TMLE is a double robust estimator.
4. TMLE it will be consistent for \(\psi(P_{0})\) is either the conditional expectation E\(_{0}\)(Y|A,W) or the exposure mechanism P\(_{0}\)(A|W) are estimated consistently. When both functions are consistently estimated, the TMLE will be efficient in that it achieves the lowest asymptotic variance among a large class of estimators. These asymptotic properties typically translate into lower bias and variance in finite samples.(Bühlmann et al., 2016)
5. The advantages of TMLE have been repeatedly demonstrated in both simulation studies and applied analyses.(Laan and Rose, 2011)
6. The procedure is available with standard software such as the tmle package in R (Gruber and Laan, 2011).

2 The G-Formula

1. \(\psi(P_{0})\,=\,\sum_{w}\,\left[\sum_{y}\,P(Y=y\mid A=1,W=w)-\,\sum_{y}\,P(Y = y\mid A=0,W=w)\right]P(W=w)\)
   where
   \(P(Y = y \mid A = a, W = w)\,=\,\frac{P(W = w, A = a, Y = y)}{\sum_{y}\,P(W = w, A = a, Y = y)}\)
   is the conditional probability distribution of Y = y, given A = a, W = w and,
   \(P(W = w)\,=\,\sum_{y,a}\,P(W = w, A = a, Y = y)\)
   
2. Using classical regression methods to control confounding requires making the assumption that the effect measure is constant across levels of confounders included in the model.
3. Alternatively, standardization allows us to obtain an unconfounded summary effect measure without requiring this assumption.The G-formula is a generalization of standardization(Greenland and Robins, 1986)

3 Causal assumptions

Under the counterfactual framework, we have to consider the following assumptions to consider the estimate of the ATE as causal:

3.1 CMI or Randomization

(\(Y_{0},Y_{1}\perp\)A|W) of the binary treatment effect (A) on the outcome (Y) given the set of observed covariates (W), where W = (W1, W2, W3, … , Wk).

3.2 Positivity

a ϵ A: P(A=a | W) > 0
P(A=1|W=w) > 0 and P(A=0| W = w) > 0 for each possible w.

3.3 Consistency or SUTVA:

The observed outcome value, under the observed treatment, is equal to the counterfactual outcome corresponding to the observed treatment for identical independent distributed (i.i.d.) variables.

4 TMLE flow chart

Source : Mark van der Laan and Sherri Rose. Targeted learning: causal inference for observational and experimental dataSpringer Series in Statistics, 2011

5 Data generation

In R we create a function to generate the data with the input number of draws and the output the observed data (ObsData) plus the counterfactuals (Y1, Y0).
The observed data:
1. Y: mortality binary indicator (1 death, 0 alive) 2. A: binary treatment for emergency presentation at cancer diagnosis (1 EP, 0 NonEP)
3. W1: Gender (1 male; 0 female)
4. W2: Age at diagnosis (0 <65; 1 >=65)
4. W3: Cancer TNM classification (scale from 1 to 4)
5. W4: Comorbidities (scale from 1 to 5)

#install.packages("broom")
options(digits=3)
generateData <- function(n){
  w1 <- rbinom(n, size=1, prob=0.5)
  w2 <- rbinom(n, size=1, prob=0.65)
  w3 <- runif(n, min=0, max=4)
  w4 <- runif(n, min=0, max=5)
  A  <- rbinom(n, size=1, prob= plogis(-0.4 + 0.2*w2 + 0.15*w3 + 0.2*w4))
  Y  <- rbinom(n, size=1, prob= plogis(-1 + A -0.1*w1 + 0.3*w2 + 0.25*w3 + 0.2*w4))
  
  # counterfactual
  Y.1 <- rbinom(n, size=1, prob= plogis(-1 + 1 -0.1*w1 + 0.3*w2 + 0.25*w3 + 0.2*w4))
  Y.0 <- rbinom(n, size=1, prob= plogis(-1 + 0 -0.1*w1 + 0.3*w2 + 0.25*w3 + 0.2*w4))
  
  # return data.frame
  data.frame(w1, w2, w3, w4, A, Y, Y.1, Y.0)
}
set.seed(7858)
ObsData <- generateData(n=1000)
True_Psi <- mean(ObsData$Y.1-ObsData$Y.0);True_Psi

[1] 0.241

Bias_Psi <- summary(lm(data=ObsData, Y~A));Bias_Psi$coef

            Estimate Std. Error t value Pr(>|t|)
(Intercept)    0.543     0.0238   22.79 6.48e-93
A              0.210     0.0300    6.99 4.98e-12

6 Data visualization

# DT table = interactive
# install.packages("DT") # install DT first
library(DT)
datatable(head(ObsData, n = nrow(ObsData)), options = list(pageLength = 5)) 

7 TMLE implementation

7.1 1st step: E\(_{0}\)(Y|A,W)

Estimation of the initial probability of the outcome (Y) given the treatment (A) and the set of covariates (W), denoted as the \(Q_{0}\)(A,W). To estimate \(Q_{0}\)(A,W) we can use a standard logistic regression model:

\(logit[P(Y=1|A,W)]\,=\,\beta_{0}\,+\,\beta_{1}A\,+\,\beta_{2}^{T}\)W.

Therefore, we can estimate the initial probability (as follows: . (1) The predicted probability can be estimated using the Super Learner library implemented in the R package “Super-Learner”6 to include any terms that are functions of A or W (e.g., polynomial terms of A and W, as well as the interaction terms of A and W, can be considered). Consequently, for each subject, the predicted probabilities for both potential outcomes and can be estimated by setting A = 0 and A = 1 for everyone respectively: and,.

8 Thank you

Thank you for participating in this tutorial.
If you have updates or changes that you would like to make, please send me a pull request. Alternatively, if you have any questions, please e-mail me.
Miguel Angel Luque Fernandez
E-mail: miguel-angel.luque at lshtm.ac.uk
Twitter @WATZILEI

9 Session Info

devtools::session_info()

10 References

Bühlmann P, Drineas P, Laan M van der, Kane M. (2016). Handbook of big data. CRC Press.

Greenland S, Robins JM. (1986). Identifiability, exchangeability, and epidemiological confounding. International journal of epidemiology 15: 413–419.

Gruber S, Laan M van der. (2011). Tmle: An r package for targeted maximum likelihood estimation. UC Berkeley Division of Biostatistics Working Paper Series.

Laan M van der, Rose S. (2011). Targeted learning: Causal inference for observational and experimental data. Springer Series in Statistics.